shows MN segment graphed on a coordinate plane. Point P lies on MN segment and is 3/4 of the way from M to N. What are the coordinates of the point P?
these are points on graph M(-4, 4), N(6,-2)
-4 to + 6 = 10
3/4 of 10 is 7.5
-4+ 7.5 = 3.5
4 to -2 is -6
3/4 of -6 is -9/2
4 - 9/2 = -1/2
(3.5, .5)
To find the coordinates of point P, we need to determine the point that is 3/4 of the way from M(-4, 4) to N(6, -2).
First, we need to calculate the differences in x-coordinates and y-coordinates between M and N:
Δx = N(x) - M(x) = 6 - (-4) = 10
Δy = N(y) - M(y) = -2 - 4 = -6
To find the new point P, we can use the formula:
P(x) = M(x) + (3/4) * Δx
P(y) = M(y) + (3/4) * Δy
Plugging in the given values and calculating, we get:
P(x) = -4 + (3/4) * 10 = -4 + 7.5 = 3.5
P(y) = 4 + (3/4) * (-6) = 4 - 4.5 = -0.5
Therefore, the coordinates of point P are (3.5, -0.5).
To find the coordinates of point P, we first need to determine the distance between points M and N. Then, we can calculate the position of point P, which is 3/4 of the way from M to N.
The distance between two points can be found using the distance formula, which is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, M(-4, 4) corresponds to x1 = -4 and y1 = 4, and N(6, -2) corresponds to x2 = 6 and y2 = -2.
Using the distance formula, we can find the distance between M and N:
d = sqrt((6 - (-4))^2 + (-2 - 4)^2)
= sqrt(10^2 + (-6)^2)
= sqrt(100 + 36)
= sqrt(136)
Now that we have the distance between M and N, we can calculate the position of point P, which is 3/4 of the way from M to N.
Using the concept of proportion, we can set up an equation to find the coordinates of point P. Let's assume the coordinates of P are (x, y).
Distance_from_M_to_P / Distance_from_M_to_N = 3/4
Using the distance formula, the equation becomes:
sqrt((x - (-4))^2 + (y - 4)^2) / sqrt(136) = 3/4
Now, we square both sides of the equation to eliminate the square roots:
((x - (-4))^2 + (y - 4)^2) / 136 = (3/4)^2
Simplifying the right side:
((x - (-4))^2 + (y - 4)^2) / 136 = 9/16
Cross-multiplying:
16((x - (-4))^2 + (y - 4)^2) = 9 * 136
Simplifying:
16((x + 4)^2 + (y - 4)^2) = 9 * 136
Now, you can solve this equation to find the values of x and y, which represent the coordinates of point P.